Pinch Points and Half-Moons in Dipolar-Octupolar Nd$_2$Hf$_2$O$_7$

While it is established that the pinch point scattering pattern in spin ice arises from an emergent Coulomb phase associated with a magnetic moment that is divergence free, more complex Hamiltonians can introduce a divergence-full part. If these two parts remain decoupled, they give rise to the coexistence of distinct features. Here, we show that the moment in Nd2Hf2O7 forms a static long-range ordered ground state, a flat, gapped pinch point excitation, and dispersive excitations. These results confirm recent theories which predict that the dispersive modes, which arise from the divergence-full moment, host a pinch point pattern of their own, observed experimentally as "half-moons. "